1963 IMO Problems/Problem 5

Problem

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$ .

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We have S = cos(π/7) - cos(2π/7) + cos(3π/7) = cos(π/7) + cos(3π/7) + cos(5π/7)

Then, by product-sum formulae, we have S * 2* sin(π/7) = sin(2π/7) + sin(4π/7) - sin(2π/7) + sin(6π/7) - sin(4π/7) = sin(6π/7) = sin(π/7)

Thus S = 1/2

1963 IMO (Problems) • Resources
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
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